Entropy – Definition and Calculations

Entropy is a very important phenomenon that explains the complexity of the universe. You may probably hear that entropy of the universe increases in all aspects. So the irregularity of the universe increases. That must be true but, we will look at this term on a much more scientific and engineering basis here. You can find lots of information and calculations about this term here.

What is Entropy?

Giving the exact meaning of the entropy without explaining it with scientific equations is very hard. The first law of thermodynamics explains the energy balance and energy conservation principles in general. And the second law of thermodynamics explains the general principles of entropy.

In the universe, there is always an entropy generation in each process. And in general, the disorder and irregularity of the universe increase. So entropy increases. We measure this disorder with the entropy increase.

For example, one may think, that in nature and on a practically daily basis, there are lots of highly ordered systems such as human bodies and cars. If the entropy increases every time, how it is possible to have these highly ordered systems? The general answer is very simple. We expense lots of kinds of power and lots of kinds of energy to produce these cars. think about all the production processes such as the casting of the engine block. The accumulated order in the engine block is much lower than the entropy generation or disorder generation in the metal casting operations. If we generalize to all the parts that make up the car, we can conclude that we created and generated a very high amount of disorder or entropy in the universe to obtain this car.

The same principle is valid for the human body. To obtain this complex and highly ordered system, there is a lot of entropy generation at the chemical reactions that created our bodies. So, entropy generation does not prevent obtaining complex structures. You just need to give more effect to obtain complex systems because of the entropy and disorder in the universe.

Molecular Disorder

We can also think the entropy as a molecular disorder of substances. So with each process, the molecular disorder of the universe increases. Because there are irreversibilities such as friction. These irreversibilities increase the entropy of the universe.

But we need a reference point to calculate the disorder. This reference point is absolute zero temperature and perfectly crystal structures. In this case, there are no uncertainties about the places and situations of the toms and molecules. We calculate the entropies of the systems according to this reference point.

This is the statement of the third law of thermodynamics. While we are making our calculations, we consider the reference point as the absolute zero and perfectly crystal structures.

But we need to state that, perfect crystal structures are important. Because, for example, amorphous structures in absolute zero temperature have uncertainty and disorder in the structure. So, their entropy is not zero.

Entropy Calculations

If we take a look at the working principles of cyclic devices in thermodynamics, the practical application of this term is very common. Because we calculate the entropy of the systems to calculate the efficiencies strictly.

Heat transfer is a very common thing in the thermodynamic calculations of systems. The energy transfer takes place as heat transfer between the environment and the parts of thermodynamical systems. So, we need to understand the entropy generation with the heat transfer to a system.

Entropy Change of Liquids and Solids

We can thermodynamically calculate the change of this property with the heat transfer or temperature change on solids and liquids. We use this equation to calculate it;

In this equation, Caverage is the average specific heat of the solid or liquid between the temperature range of T2 and T1. So, it is very simple to calculate the change of this physical property for solids and liquids.

Entropy Change of Ideal Gases

Ideal gases are very important in thermodynamic calculations. We use them as a very good approximation of lots of kinds of systems. The calculations with ideal gases are much easier in engineering systems. And they provide extra time with minimal deflections in calculations.

We can calculate the entropy change of the ideal gases easily. We can make our calculations for both the constant pressure and the constant volume. Because the specific heat values change with their changing values of them. So, you can use these equations respectively.

For constant pressure processes;

In this equation, R is the ideal gas constant which is 8.314J/K⋅mol. v2 is the pressure of the second state and P1 is the pressure of the first state. Cp, the average is the average specific heat in constant pressure processes.

And also, for constant volume processes;

In this equation also, the Cv, the average is the average specific heat in constant volume processes between the temperature range between T2 and T1. And V2 is the specific volume of the second state. And V1 is the specific volume of the first state.

As you see in the two equations above, with the increasing specific heat, the total entropy generation increases. Also, there is a direct relationship between the increasing differences between pressures, volumes, and temperatures of the first and second situations, and the entropy generation increases.

Entropy Generation with Heat Transfer

In the net entropy change calculations of the thermodynamical systems, we use this equation for the internally reversible systems in which there is no temperature change. So, this will be very useful in the calculation of the thermodynamic systems that work in the constant temperature environment. For example, a compressor of a refrigerator works at an ambient temperature that does not change with the heat dissipation of the compressor.

So, we use this equation to calculate the entropy generation in heat transfer;

In this equation, the total change of the entropy is equal to the division of Q with the ambient temperature.

Q is the total heat energy that is transferred between the irreversible system and the environment. T0 is the constant temperature of the environment which does not change with the heat transfer.

The change of entropy depends on the direction of the heat transfer. If the heat transfer is from the system to the environment, it will take a negative sign. So the entropy of the system will decrease, but the entropy of the environment increases. And also if heat transfer is from the environment to the system, it will take a positive sign. So, the entropy of the system increases and the environment decreases.

Increase of Entropy Principle

We stated above that entropy always increases in the universe and all of the processes take place. So, we can calculate the total increase of entropy in all types of systems.

For example, there is a heat transfer from 800K heat source to 500K heat source. And the total heat transfer between these heat sources is 100Kj. So, according to the equation above, we can calculate the entropy changes of the systems above;

S1 = -100/800 = -1/8 Kj/K

S2 = 100/500 = 1/5 Kj /K

And if we calculate the total entropy of the system we can find;

Total Entropy = -1/8 + 1/5 = 0.075Kj /K

As you see above, there is an entropy generation in each heat transfer process. So, we can calculate the total entropy change of the systems with the calculations by summing all the entropy changes of the environments.

Irreversibilities

Irreversibilities are the main cause of entropy generation. In all kinds of systems, there are irreveisrsibilities. We can not escape from irreversibilities in the actual processes. So, entropy generation gives a very important insight into the level of irreversibilities of the systems.

Because irreversibilities are decreasing the efficiencies of the systems. If we decrease the irreversibilities, the system will be more efficient. And if the entropy generation is higher, the irreversibilities in a system are higher.

Engineers and designers are using entropy generation to measure the level of irreversibilities.

There are different causes of the irreversibilities. Around them, heat leakages, friction, and other chaotic physical things. We can not produce a piston-cylinder system that has zero friction and is completely insulated. So, there is always entropy generation in these systems.

For reversible processes, the entropy generation is zero. Because there are no irreversibilities in the processes. But we can not obtain a system in which the total entropy generation is negative. So, this is the general rule of the universe.

Direction of Processes

The entropy generation is about the direction of the processes. There are no kinds of heat transfer from 100K temperature to 200K temperature. Because there must be a positive temperature difference for the heat transfer. If there is a heat transfer from 100K to 200K in nature, the total entropy will be negative. So, we can say that processes need to occur in the direction where there is an entropy increment.

You can understand that the general working principle of this phenomenon is simply like this.

Conservation of Entropy?

Unlike energy, entropy is a non-conserved physical property. There is always a generation of it. And always it is possible in terms of the bigger picture. If we would like to say that entropy is conserved, we need to obtain reversible systems. In the practical world, there is no place for reversible processes.

In engineering, we do not want to deal with irreversibilities. Because the presence of irreversibilities means low efficiency.

Isentropic Processes

In isentropic processes, the total change of the systems is zero. So, in isentropic processes, there are no irreversibilities to increase entropy. In thermodynamical calculations, we use isentropic processes in general. This is because the isentropic processes are very important in the calculation of the total irreversibilities of the systems.

If we consider the thermodynamical applications as isentropic processes, we consider that there are no irreversibilities. This is because there is no change of entropy because of the irreversibilities. So, we know the maximum efficiencies that our systems can attain with his approach. And how much the actual processes are converging to these processes.

The entropy of Water and Steam Systems

There are lots of kinds of thermodynamical systems that work with pure water as a primary fluid. There are lots of phases and temperature changes that occur in water to obtain useful mechanical work or energy from these systems. The biggest example of these systems is the steam turbines. Water undergoes lots of kinds of phases changes in steam turbines.

There are also phase changes for these systems. So, we need to calculate the entropy change with a different approach than the solids, liquids, and ideal gases.

So, if we want to calculate the total entropy change of these systems, we need to calculate the entropy change of the water over the different periods. For water and refrigerant fluids, there are formulations to calculate the entropy according to pressure and temperature. But these formulations are very complex to calculate easily. We need some computational resources to calculate them.

We have thermodynamical tables where we can read these values of these substances per kilogram basis.

Entropy in T-s Diagrams

From T-s diagrams, we can conclude about the entropy and phase change situations of water and other substances.

You can see the line on the T-s diagram or temperature entropy diagram of a pure substance such as water.

The left branch of the T-s diagram is the saturated liquid line that shows the starting point of the phase change for changing temperatures. Also, the right branch of the line is the saturated vapor line that gives the phase change to complete superheated vapor.

So, the section on the left side of the left branch is the compressed liquid which is the liquid phase of the substance. And also the section between the branches is the saturated-liquid-vapor mixture section.

The temperature and pressure of a pure substance change according to these diagrams. At a specific point, we can read the entropy per kg values from this diagram for a pure substance.

There are thermodynamical tables where you can easily read the total entropy per kilogram according to the changing temperatures and pressures.

Saturated Liquid Vapor Mixture

This mixture is the phase change of a pure substance from liquid to vapor. This phase change occurs in most engineering and thermodynamical systems. So, we can calculate the entropy of a substance that is in a saturated liquid-vapor mixture with simple mathematical extrapolations.

For example, at a specific temperature, we would like to calculate the total entropy of a substance that has 60% of vapor and 40% liquid in it. So the vapor mixture percentage of 60% or 0.6 that we will use in our calculations. The entrpy per kilogram is;

In this equation, Sf is the entropy per kilogram of the substance that is saturated liquid and Sg is the entropy per kilogram of the saturated vapor at a specific temperature. So if you know the percentage, place it into the X. And you can find of the liquid-vapor mixture.

If you multiply this value, you can find the total entropy of the substance that you have.

So in the total entropy calculations of the phase changes of the substances, you can use these values in different phases. And you can calculate the total entropy change of a total system that works with a primary fluid.

Isentropic Efficiency Calculations

We stated that, if there are no irreversibilities, the entropy change will be zero for the system. So, we call these systems isentropic systems. In engineering calculations, we use the isentropic efficiency calculations to see how many irreversibilities the actual systems have. We can make out calculations for different kinds of engineering systems.

Turbines

Turbines are very important systems in engineering. We use them to produce shaft work from the flow energy. The fluid flows through the turbine wings, and the turbine rotates. The output rotating shaft produces the mechanical work.

The inlet and exit pressure of the fluid is an important phenomenon in turbines. We calculate the total shaft energy by calculating the difference between the input and output pressures of the fluids.

We can calculate the isentropic efficiencies of turbines by dividing the actual work output by the work output if the pressure drop of the fluid is isentropic.

Compressors and Pumps

Compressors and pumps are also very important devices to increase the pressure of fluids. We use compressors for compressible fluids such as gases, and pumps for incompressible fluids such as water.

In their systems, there is a work input through a shaft, and there are vanes that take the low-pressure fluids to increase their pressure. So, their working principles are reversed of them.

If you want to calculate the isentropic efficiency of the compressors and pumps, you need to divide the total actual work input of the shaft by the work input to pressurize the liquid without change. So, you need to consider no entropy change in the two pressure states of the working fluid.

Nozzles

We use nozzles in engineering systems where we need to change the kinetic energy of the flowing fluids. So, it is very important to have nozzles in the engineering applications such as refrigerators.

So, to calculate the isentropic efficiency of nozzles, you need to divide the actual kinetic energy change of the fluid by the kinetic energy change of the fluid without any entropy change. And you will calculate the total kinetic energy change without the irreversibilities of the nozzle.

Entropy Transfer

If there are heat and mass transfers between systems, we need to consider the entropy transfer with them. In the calculation of the total change of the entropy, we need to include the entropy transfer through the boundaries.

Entropy transfer with heat transfer;

S = Q/T

With mass flow;

S = ms

And we know the general entropy generation for the systems. So, we can calculate the total entropy change for these systems.

We can calculate the Sin and Sout with the consideration of mass flow and heat transfer entropy transfer. And we explained the generation of it though the processes. So the total change of the entropy of a system is like that.

Concluding Remarks on Entropy

These are the general statements about entropy. It is a very important physical factor that we use in engineering. There are lots of kinds of complex calculations but the main logic is very simple. It is just the total disorder of the systems.

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